Find the critical numbers of the function. (Enter your answers as a comma-separated list. Use n to denote any arbitrary integer values. If an answer does not exist, enter DNE.) f(θ)=6cosθ+3sin2θ g
Answers
Answer:
The critical value of \(f(\theta) = 6\cdot \cos \theta + 3\cdot \sin 2\theta\) are given by \(\theta \approx 0.091\pi \pm 2\pi\cdot n\) or \(\theta \approx 0.909\pi \pm 2\pi \cdot n\), \(\forall \,n \in \mathbb{N}\)
Step-by-step explanation:
The function to be evaluated is \(f(\theta) = 6\cdot \cos \theta + 3\cdot \sin 2\theta\), the first derivative of the function must be taken in order to determine the set of critical numbers. Each derivative are found by using the differentiation rule for a sum of functions and rule of chain and subsequently simplified by trigonometric and algebraic means:
First derivative
\(f'(\theta) = - 6 \cdot \sin \theta +6\cdot \cos 2\theta\)
\(f'(\theta) = -6\cdot \sin \theta + 2\cdot (\cos^{2}\theta-\sin^{2}\theta)\)
\(f'(\theta) = -6\cdot \sin \theta + 2\cdot [(1-\sin^{2}\theta-\sin^{2}\theta)]\)
\(f'(\theta) = -6\cdot \sin \theta + 2\cdot (1-2\cdot \sin^{2}\theta)\)
\(f'(\theta) = -6\cdot \sin \theta + 2 - 4\cdot \sin^{2}\theta\)
\(f'(\theta) = -4\cdot \sin^{2}\theta - 6\cdot \sin \theta +2\)
The procedure to determine the critical number of the given function are described briefly:
1) First derivative is equalised to zero.
2) The resultant equation is solved.
Then,
\(-4\cdot \sin^{2}\theta - 6\cdot \sin \theta +2 = 0\)
Whose roots are:
\(\sin \theta_{1} \approx 0.281\) and \(\sin \theta_{2} \approx -1.781\)
The sine function is a continuous function with a range between 1 and -1, so, only the first root offers a realistic solution. In addition, such function is positive at first and second quadrants and has a periodicity of \(2\pi\) radians, the family of critical values are determined by the unse of inverse trigonometric functions:
\(\theta \approx \sin^{-1} 0.281\)
There are two subsets of solutions:
\(\theta \approx 0.091\pi \pm 2\pi\cdot n\) or \(\theta \approx 0.909\pi \pm 2\pi \cdot n\), \(\forall \,n \in \mathbb{N}\)
Related Questions
-2x (-3x²y)(2y) = _______.
Answers
Answer:
12x³y²
Step-by-step explanation:
You want the product -2x (-3x²y)(2y).
ExponentAn exponent is a means of telling the number of times the base is a factor in a product. For example, the exponent 2 means the base is a factor twice in the product:
x² = x·x
ApplicationWhen the same factor appears a number of times in a product, an exponent can be used to represent that number.
(-2x)(-3x²y)(2y) = (-2)(-3)(2)(x·x²)(y·y)
The factor x appears 3 times in the product, and the factor y appears twice. The numerical product can be simplified, so the result is ...
12x³y²
find the value of x
Answers
Answer:
31.8 that is it I just need to fill a 20 character long answer
answer ASAP pls
thanks and God bless you
Answers
We need to plug in the numbers for x and y into the expression for x and y.
\(2x^{2} - y\\\\2(3)^{2} - (-2)\\\\2(3)^{2} + 2\\\\18 + 2\\\\20\)
The simplified answer is 20.
PLEASE HELP !!!!!!!!
Give me two rules of exponents that you have learned so far
in middle school maths, any basic rule would help thanks :)
Answers
Answer: I learned math
Step-by-step explanation:
Answer:
Step-by-step explanation:
Multiplication:
Suppose we decide to use the base b and that we multiply b^a and b^c together. The relevant rule of exponents tells us that the product is
b^(a + c). In words, we add together the two exponents a and c.
Exponentiation:
Suppose we have b^c and wish to raise all of b^c to the power d. The base remains the same: It is b. Starting with b^c, we enclose b^c inside parentheses and then raise the quantity inside parentheses to the power d:
(b^c)^d. Note that this simplifies to b^(cd), in which we multiply the powers together.
A baker makes 8 apple pies. Each apple pie weighs 6 ounces.
How many total pounds of apple pies does the baker make?
Answers
Answer: 3 pounds of apple pie
Step-by-step explanation:
If there are 8 pies and each one weighs 6 ounces, then we can multiply 6 and 8 to get a total of 48 ounces. Then we would convert ounces to pounds. 48 ounces is 3 pounds.
7x + 8y = 25 9x + 10y =35 elimination method please
Answers
\(7x +8y = 25 ~~~....(i)\\\\9x +10y = 35~~~ ....(ii)\\\\\text{Multiply equation (ii) by}~ \dfrac 79 ~ \text{and do (ii) - (i):}\\\\\dfrac 79 (9x +10y) - (7x +8y) = \dfrac 79 (35) - 25\\\\\\\implies 7x +\dfrac{70y}9 - 7x -8y = \dfrac{245}9 -25\\\\\\\implies - \dfrac{2y}9 = \dfrac{20}9\\\\\\\implies -2y =20\\\\\implies y =-10\\\\\text{Substitute}~ y = -10~ \text{in equation (i):}\\\\\\7x +8(-10) = 25\\\\\implies 7x = 25 +80\\\\\implies 7x = 105 \\\\\implies x =15\\\\\)
\(\\\text{Hence}~ (x,y) = (15,-10)\)
The outdoor temperature is 4 degrees below 0, or
.
The indoor temperature is 72 degrees above 0, or
.
Answers
Hi Chloe!
Answer:
The outdoor temperature is -4 degrees.
The indoor temperature is 72 degrees.
Explanation:
We use negative integer to represent loss, below , or withdrawal etc. On the other hand we use positive integers to show gain , above , deposit etc.
your base salary is $8.00. For every bag of trash you to take out you get $2.00. A. write an equation to represent how much money you can make? B. If you take out 12 bags of trash how money do you make?
Answers
HURRY!!!!!!!!!!!!!!!
The table shows the proportional relationship between the number of tickets required per game at a carnival. Games 4 7 10 Tickets 20 35 50 Determine the constant of proportionality.
Correct answer will get brainiest
Answers
Answer:
Therefore, the constant of proportionality for the given proportional relationship is 5.
Step-by-step explanation:
To determine the constant of proportionality in the given proportional relationship, we can use the formula:
Constant of Proportionality = Tickets / Games
Let's plug in the values from the table:
For the first set of values (4 games and 20 tickets):
Constant of Proportionality = 20 / 4 = 5
For the second set of values (7 games and 35 tickets):
Constant of Proportionality = 35 / 7 = 5
For the third set of values (10 games and 50 tickets):
Constant of Proportionality = 50 / 10 = 5
In all three cases, the constant of proportionality is equal to 5.
Therefore, the constant of proportionality for the given proportional relationship is 5.
17 The table below shows the distance a car has traveled.
50
20
f
40
Minutes
Distance
Traveled
(in miles)
What is the meaning of the slope of the linear model for the data?
60
100
a) The car travels 5 miles every minute.
b) The car travels 4 miles every minute.
c) The car travels 4 miles every 5 minutes.
d) The car travels 5 miles every 4 minutes.
125
80
100
Answers
Given statement solution is :- None of the given options (a, b, c, or d) match the meaning of the slope. The correct interpretation is that the car travels approximately 0.8 miles every minute.
To determine the meaning of the slope of the linear model for the given data, let's analyze the information provided. The table represents the distance traveled by a car at different time intervals.
Minutes | Distance Traveled (in miles)
50 | 20
20 | f
40 | 60
100 | a
125 | 80
100 | 100
To find the slope of the linear model, we need to calculate the change in distance divided by the change in time. Let's consider the intervals where the time changes by a fixed amount:
Between 50 minutes and 20 minutes: The distance changes from 20 miles to 'f' miles. We don't have the exact value of 'f', so we can't calculate the slope for this interval.
Between 20 minutes and 40 minutes: The distance changes from 'f' miles to 60 miles. Again, without knowing the value of 'f', we can't calculate the slope for this interval.
Between 40 minutes and 100 minutes: The distance changes from 60 miles to 'a' miles. We don't have the exact value of 'a', so we can't calculate the slope for this interval.
Between 100 minutes and 125 minutes: The distance changes from 'a' miles to 80 miles. Since we still don't have the exact value of 'a', we can't calculate the slope for this interval.
Between 125 minutes and 100 minutes: The distance changes from 80 miles to 100 miles. The time interval is 25 minutes, and the distance change is 100 - 80 = 20 miles.
Therefore, based on the given data, we can conclude that the car travels 20 miles in 25 minutes. To determine the meaning of the slope, we divide the distance change by the time change:
Slope = Distance Change / Time Change
= 20 miles / 25 minutes
= 0.8 miles per minute
So, none of the given options (a, b, c, or d) match the meaning of the slope. The correct interpretation is that the car travels approximately 0.8 miles every minute.
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brainliest points whatever you want please
Answers
Annual rate of return earned when the share was sold at $50 per share is 20%
Annual rate of return earned when the share was sold at $60 per share is 40%
How to calculate annual rate of returnAnnual rate of return is given by the formula
let end of the year price = eyp
let beginning of the year price = byp
= {(eyp - byp) / byp} * 100
Annual rate of return at share price of $50 per share
when there is paying of dividends the value is added to the the eyp
eyp = 50 * 100 + 2 * 100 * 5 = 6000
byp = 50 * 100 = 5000
= {(6000 - 5000) / 5000} * 100
= 20%
Annual rate of return at share price of $60 per share
eyp = 60 * 100 + 2 * 100 * 5 = 7000
byp = 50 * 100 = 5000
= {(7000 - 5000) / 5000} * 100
= 40%
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one card is drawn from a pack of 52cards each of the 52 cards being equally likely to be drawn. what is the probability that the card drawn is a king?
Answers
The probability of drawing a king from a standard deck of 52 cards is 1/13.
In a standard deck of 52 playing cards, there are four kings: the king of hearts, the king of diamonds, the king of clubs, and the king of spades.
To find the probability of drawing a king, we need to determine the ratio of favorable outcomes (drawing a king) to the total number of possible outcomes (drawing any card from the deck).
The total number of possible outcomes is 52 because there are 52 cards in the deck.
The favorable outcomes, in this case, are the four kings.
Therefore, the probability of drawing a king is given by:
Probability = (Number of favorable outcomes) / (Number of possible outcomes)
= 4 / 52
= 1 / 13
Thus, the probability of drawing a king from a standard deck of 52 cards is 1/13.
This means that out of every 13 cards drawn, on average, one of them will be a king.
It is important to note that the probability of drawing a king remains the same regardless of any previous cards that have been drawn or any other factors.
Each draw is independent, and the probability of drawing a king is constant.
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will give brainliest
Answers
Answer:
About 8.2 cm
Step-by-step explanation:
Assuming that a is one of the legs of this right triangle, then using the Pythagorean Theorem:
\(a=\sqrt{15.3^2-12.9^2}\approx 8.2\)
Hope this helps!
Which of the following are the correct properties of slope?
Check all that apply.
O A. The slope of a line that moves up from left to right is positive.
a
B. The slope of a line is always positive.
C. Some lines have an undefined slope.
D. A steep line has negative slope.
E. If two lines have the same slope, then they are the same line.
Answers
Answer:
I think one of them may be C and B but I'm not sure
Vincent bought a TV for $467.00. Sales tax is 8.7%. How much money will Vincent have to pay in sales tax?
Question 8 options:
$0.41
$4.06
$406.29
$40.63
Answers
Answer:
40.63
Step-by-step explanation:
8.7% of 467 is 40.629 but rounds to 40.63
Answer:
D. $40.63
Step-by-step explanation:
8.7% of $467 = 0.087 × 467 = $40.629
40.629 rounds up to $40.63
You sailed 0.055 units to the left and found treasure at 0.085 units find where the ship started
Answers
The ship started at the Position 0.14 units.
To determine the starting position of the ship, we can consider the distance sailed to the left and the distance to the treasure.
Given that you sailed 0.055 units to the left and found the treasure at 0.085 units, we can represent this situation mathematically as follows:
Starting position + Distance sailed to the left = Distance to the treasure
Let's assign a variable, "x," to represent the starting position of the ship.
The equation becomes:
x - 0.055 = 0.085
To find the value of x, we can solve this equation by isolating x on one side:
x = 0.085 + 0.055
x = 0.14
Therefore, the ship started at the position 0.14 units.
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More than two-thirds of undergraduate students who graduated with a bachelor's degree had student loan debt. The average student loan debt among these graduating seniors was $28,654. The average interest rate on student loans was 6.1%. How much interest did a student with $28,654 in student loan debt pay in the first year? Round to the nearest cent.
Answers
The interest paid for the loan is $1748
Given that there is a 6.1% interest rate for a loan of $28,654 in student loan debt pay in the first year,
We need to find the interest paid for the same.
So,
Simple Interest = principal × time × rate / 100
= 28654 × 1 × 6.1 / 100
= 28654 × 0.061
= 1747.894
= 1748
Hence the interest paid for the loan is $1748
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Hmmm , can someone help with this one?
Answers
Answer:
answer choice c
Step-by-step explanation:
A local theatre sells out for their show. They sell all 500 tickets for a total purse of $7,860.00. The tickets were priced at $15 for students, $12 for children, and $18 for adults. If the band sold three times as many adult tickets as children's tickets, how many of each type were sold?
Answers
The number of each type of tickets sold are;
180 adults tickets
260 students tickets
60 children tickets
How to solve simultaneous equation word problems?
Let us solve to get the solution using 3 variables.
Let c be the number of children's tickets
Let s be the number of student tickets
Let a be the number of adult tickets
We are told that the total number of tickets was 500
Thus;
a + s + c = 500 ------(1)
The number of adult tickets was 3 times the number of children's tickets. Thus;
a = 3c ------(2)
The total cost of the tickets was $8070. Thus;
18a + 15s + 12c = 7860 ----(3)
We conclude that solving these three equations simultaneously gives us;
a = 180 adults
s = 260 students
c = 60 children
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Seven more than twice a number is less than negative thirteen. Find all numbers that make this statement true. The answer is (Type an inequality using x as the variable.)
Answers
Answer:
x < -10
Explanation:
First, we need to write the sentence seven more than twice a number is less than negative thirteen as an algebraic form. So, we can represent each part of the sentence as:
Seven: 7
more: +
Twice a number: 2x
is less than: <
negative thirteen: -13
So, the inequality that represents the sentence is:
7 + 2x < -13
Now, we need to solve for x, so subtract 7 from both sides:
7 + 2x - 7 < -13 - 7
2x < -20
Divide both sides by 2:
2x/2 < -20/2
x < -10
Therefore, the numbers that make the statement true are the numbers that are less than - 10.
So, the answer is:
x < -10
Help asap
in each of the following replace the * with the smallest digit to make it divisble by 11
7,01,69,30*
Answers
The smallest number that will make it divisible by 11 is n = 2 and the number is A = 7,01,69,302
Given data ,
Let the missing number be represented as n
Now , the value of the original number is A = 7,01,69,30*
where * represents = n
To make a number divisible by 11, the difference between the sum of the digits in the odd-numbered positions and the sum of the digits in the even-numbered positions must be a multiple of 11.
On simplifying the equation , we get
To make the difference between the sums a multiple of 11, we need to add the smallest digit to the end. The smallest digit is 8
Hence , the number that is divisible by 11 is 7,01,69,302
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Sara took a large bag of flour and put the same amount into four different containers she had 1.7 lb of flour remaining if she started 16.1 lb of flour how much flour is in each container
Answers
Answer:
3.6 pounds of flour per container.
Step-by-step explanation:
Subtract 1.7 by the starting amount, then divide that number by 4.
16.1 - 1.7 = 14.4
14.4/4 = 3.6
Answer:
3.6 lb
Step-by-step explanation:
the numbers of flour in each container =
(16.1 - 1.7)/4 = (14.4)/4 = 3.6 lb
The decimal -2.97 is the solution to which subtraction problem?
-6.1 - 3.13
-6.1 - (-31.3)
-0.61 - (-3.13)
-6.1 - (-3.13)
-6.1- 3.13 -6.1 - (-3.13) -0.61 - (3.13) -6.1 - (3.13)
Answers
Answer:
Option 4: -6.1 - (-3.13)
Step-by-step explanation:
Let's solve each option individually to see if -2.97 fits as the solution:
-6.1 - 3.13 = -9.23
-6.1 - (-31.3) = 25.2
-0.61 - (-3.13) = 2.52
-6.1 - (-3.13) = -2.97
As you can see, -2.97 is the answer to Option 4.
So, the decimal -2.97 is the solution to the subtraction problem in Option 4.
Answer:
Its D
Step-by-step explanation:
Got it right brother
Find the equation of this line.
[?]
y =
x + [ ]
![Find the equation of this line.[?]y =x + [ ]](https://i5t5.c14.e2-1.dev/h-images-qa/contents/attachments/pBxDQsPpbbqoS2Y1k6cDkE2NBneIgcCt.png)
Answers
the slope intercept form is y = mx + b
b is the y-intercept
m is the slope
use rise over run to find the slope
solve the following equation algebraically ? please !
Answers
Answer: A
Step-by-step explanation:
Answer:
The answer is B.
Step-by-step explanation:
The opposite of squared is square root, so you'll have to square root 192 and you'll get the answer 13.86.
The reason it is B, because the x could have been negative, when you're squaring a negative you get a positive. that's why B is the answer because we do not know if it's the negative or positive.
(WILL MAKE BRAINLIST)
give the points (4,-5) and (0,-2) on a line find the values of M and B.
y= mx+b
M= ?
B= ?
enter answers as integers and or as reduced fractions in the form A/B
Answers
Answer:
m = -3/4; b = -2
Step-by-step explanation:
y = mx + b
First, we use the coordinates of the points to find the slope, m.
m = (y2 - y1)/(x2 - x1) = (-2 - (-5))/(0 - 4) = (-2 + 5)/(-4) = -3/4
The slope, m = -3/4.
Now we use the slope and the coordinates of one point in the equation y = mx + b to find b.
y = mx + b
y = (-3/4)x + b
Let's use point (0, -2).
-2 = (-3/4)(0) + b
b = -2
Answer: m = -3/4; b = -2
Use the average daily balance method to compute the finance charge on the credit card account
for the month of August (31 days). The starting balance from the previous month is $270. The
transactions on the account for the month are given in the table to the right. Assume an annual
interest rate of 19% on the account and that the billing date is August 1st.
aug 7th. payment of $70
aug. 10 purchase of $135
aug. 15 purchase of $20
aug. 29 purchase of $25
what is the finance charge for the month of August?
Answers
The finance charge for the month of August is $4.93.
What is a percentage?A ratio or value that may be stated as a fraction of 100 is called a percentage. Moreover, it is indicated by the symbol "%."
Given:
The starting balance from the previous month is $270.
The transactions on the account for the month are given in the table to the right.
Assume an annual interest rate of 19% on the account and that the billing date is August 1st.
After finding the balance outstanding each day, we sum it up.
And get the average as $305.4839
Now we find the average balance by dividing it by 31 days of the August month.
Average balance = $305.4839
Finance charge = $305.4839 x19 x 31/365 days
Finance charge = $4.929589
Finance charge = $4.93
Therefore, the Finance charge = $4.93.
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A pound of ground beef costs $5. what is the cost of 1/2 pound?
Answers
Answer:
$2.50
Step-by-step explanation:
x/5 = 1/2
x is equal to 1 because it is half of the total and 1 is half of 2.
5 is equal to 2 because it is the whole amount and 2 is also the whole amount.
If you then solve for x by doing
5(1) = 5
5/2 = 2.5
You get 2.5 or $2.50.
Answer:
simple $2.50
Step-by-step explanation:
1 pound is $5 so half of that is $2.50
Is this correct?
Or please provide an explanation.
Answers
Answer:
The answer selected is correct
Step-by-step explanation:
When talking about balance, having -X (being X in the negative) , means owing X.
Dylan owes $19.25
Elise owes $42.75
Francesca owes $23
Jamaal owes $35.50
So naturally, Dylan owes the least.
is y=-7/9x+2 proportional or non proportional